Analytic Solution for Subharmonic Oscillation

In a step-down converter with a fixed frequency and with peak current control, the inductor current flows continuously, and it is well known that if the duty cycle is 50% or higher, subharmonic oscillations occur. In nearly all recent ICs for use in power supplies, a slope compensation circuit is provided to address this issue.

This time, we stray somewhat from derivation of transfer functions for DC-DC converters to explain an analytic solution for subharmonic oscillation, an understanding of which is important.

Fig.10

The coil (inductor) current waveform in a current-mode step-down converter is similar to that in Fig. 10. Here the current value at a given time is I_n, and the current value one period later is I_n+1.

Further, the turn-on time is t_ON(n), the turn-off time is t_OFF(n), the slope of the coil current while turned on is m₁, and the slope of the coil current while turned off is m₂.

These can be used to express I_n+1 in terms of I_n as in equation 3-15 below.

Here, m₁ and m₂ can be expressed by equations 3-16 and 3-17 respectively.

Fig. 11 below shows a PWM input waveform corresponding to Fig. 10. If the current sense gain is R_S, then immediately after turning on the PWM input is R_SI_n.

Fig.11

Further, the current that has increased during t_ON becomes m₁t_ON(n), and so the PWM input peak voltage V_C is as expressed by equation 3-18.

Also, ｔ_OFF(n) in equation 3-15 can be rewritten as in equation 3-19 below. Here T is the period.

From equation 3-18, t_ON(n) is found, and upon substitution into equation 3-15 and calculating, equation 3-20 is obtained.

｛I_n+1-I_n} is a geometric progression, and so the condition for not causing subharmonic oscillation is that this geometric progression converges to 0 as n → ∞. In other words, the condition is expressed by equation 3-21 below.

Substituting equations 3-16 and 3-17 into 3-21, we obtain the following conditional formula 3-22.

Thus the condition for not causing subharmonic oscillation is that the duty cycle D be 1/2, that is, 50% or lower. Referring to our statement at the outset, then, we have derived the analytic solution indicating that at a duty cycle above 50%, subharmonic oscillations occur.